Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The same polynomial has four zeros in the annulus 1 < |z| < 2

Example

The polynomial z5+3z+1 has exactly four zeros in the annulus 1<∣z∣<2.

Facts & Assumptions

Given: The polynomial p(z)=z5+3z+1.

[L1]

Rouché's theorem preserves the zero count on a circle (Rouche's theorem in the classical strict-inequality form).

Verification

technique · direct
1.1L1givenalgebra

On the circle ∣z∣=2, ∣3z+1∣≤3∣z∣+1=7<32=∣z5∣. So [L1] applied to p and z5 shows that p has five zeros in ∣z∣<2, counted with multiplicity.

1.2L1givenalgebra

On the circle ∣z∣=1, ∣z5+1∣≤2<3=∣3z∣. Another application of [L1] shows that p has exactly one zero in ∣z∣<1.

2.1step 1.1step 1.2∎

Therefore the number of zeros in the annulus 1<∣z∣<2 is 5−1=4.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources